There are many ways to numerically represent of chemical systems in order to compute their electronic structure. Basis functions may be localized in real-space (atomic orbitals), in momentum-space (plane waves), or in both components of phase-space. Such phase-space localized basis functions in the form of wavelets, have been used for many years in electronic structure. In this paper, we turn to a phase-space localized basis set first introduced by K. G. Wilson. We provide the first full study of this basis and its numerical implementation. To calculate electronic energies of a variety of small molecules and states, we utilize the sum-of-products form, Gaussian quadratures, and introduce methods for selecting sample points from a grid of phase-space localized Wilson basis. Both full configuration interaction and Hartree-Fock implementations are discussed and implemented numerically. As with many grid based methods, describing both tightly bound and diffuse orbitals is challenging so we have considered augmenting the Wilson basis set as projected Slater-type orbitals. We have also compared the Wilson basis set against the recently introduced wavelet transformed Gaussians (gausslets). Throughout, we give comments on the implementation and use small atoms and molecules to illustrate convergence properties of the Wilson basis.
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